
Symmetry in Nonlinear Mathematical Physics  2009
Dmytro Popovych (National Taras Shevchenko University of Kyiv, Ukraine)
Generalized IWcontractions of Lie algebras
Abstract:
We prove that there exists just one pair of complex fourdimensional Lie algebras
such that a welldefined contraction among them is not equivalent to a generalized IWcontraction
(or to a oneparametric subgroup degeneration in conventional algebraic terms).
Over the field of real numbers, this pair of algebras is split into two pairs with the same contracted algebra.
The example we constructed demonstrates that even in the dimension four generalized IWcontractions
are not sufficient for realizing all possible contractions,
and this is the lowest dimension in which generalized IWcontractions are not universal.
Moreover, this is also the first example of nonexistence of generalized IWcontraction
for the case when the contracted algebra is not characteristically nilpotent
and, therefore, admits nontrivial diagonal derivations.
The lower bound (equal to three) of nonnegative integer parameter exponents which are sufficient
to realize all generalized IWcontractions of fourdimensional Lie algebras is found.
We also present a simple and rigorous proof of the known claim
that any diagonal contraction (e.g., a generalized IWcontraction) is equivalent to
a generalized IWcontraction with integer parameter powers.
Presentation

